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Fangflora3
16.10.2019 •
Mathematics
Find the area of the largest rectangle (with sides parallel to the coordinate axes) that can be inscribed in the region enclosed by the graphs f(x) = 18 - and g(x) = 2 - 9 by answering the following:
a. describe the width of the rectangle in terms of x
b. describe the height of the rectangle in terms of f(x) and g(x)
c. describe the height of the rectangle in terms of x
d. write the function a(x) of the area of the rectangle in terms of the width and height found in parts a& c.
e. find the maximum of a
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Ответ:
ιHi
We know that the length of the vertical side of the rectangle is the distance between the two function.
So we have l = f(x) - g(x)
Don't forget that we have two of these sides at two different x values.
We don't want to get confuse so lets call the left side of the y axis x1 and x2 for the right axis. ( when you're writing it put the x bigger than 1 or 2)
So we have L1 = f(x1) -g(x1)
and L2 = f(x2)-g(x2)
Before we start solving, we need to remember that there is no Y value inside the graphs. We have the others sides W, each of them intersect one function twice.
w1 = w2 = x2 - x1
But we know that f and g are symmetric across y axis, for that reason x1= -x2
Hint: Symmetric is when one shape becomes exactly like another if you flip, slide or turn it. (Google definition)
Let's go back to work
Now we have w=-x1 -x1 = -2x1
Now plug L1 and W into the area equation
A = L * W = (f(x1) - g(x1)) * (-2x1)
Now replace their values
A(x) = ((18-x²) - (2x²-9)) * (-2x)
A(x) = -2x (27-3x²)
A(x) = 6x³ -54x
We need to optimize this function. How?
Take the derivative and find the Zeros
A(x) = 18x² -54 = 0
x² = 3
x = +/- √3
Now we have the negative value, which is x
Plug it into the area of function
A(-√3)= 6(-√3)³ -54 (-√3)
= 54√3 - 18√3
= 36√3
= 62.35
This is the final answer.
I hope that's help.. if not I am sorry.
I don't solved by myself, my friends help me too. So that's why it took me so long to answer.
Ответ:
B
Step-by-step explanation:
easy
2x5 is 10, 3x5 is 15
15+10 = 25
25-5 = 20
4x5=20